Expected Number of Cards Turned Over Until the First Ace

A standard 52-card deck contains 4 aces. You reveal cards one by one and stop at the first ace. What is the expected number of cards revealed?

Answer

Expected number = 53 / 5 = 10.6

So on average, you will turn over 10.6 cards before seeing the first ace.

Key idea: Think in terms of gaps

Imagine placing the 4 aces into the deck. These 4 aces split the deck into 5 gaps:

before
1st ace
A
between
A
between
A
between
A
after
last ace

The remaining 48 non-ace cards are distributed among these 5 gaps. By symmetry, each gap has the same expected number of cards.

Expected non-aces before first ace = 48 / 5 = 9.6

Then we add 1 more card for the first ace itself:

Expected position of first ace = 9.6 + 1 = 10.6

Short derivation summary

  1. There are 4 aces, so they create 5 possible gaps for non-ace cards.
  2. The 48 non-aces are symmetrically spread across these 5 gaps.
  3. So the expected number of non-aces before the first ace is 48/5.
  4. Add 1 for the first ace itself.
  5. Therefore, the expected number of cards turned over is 48/5 + 1 = 53/5 = 10.6.